The thickness of the membrane is d, which can vary over the element. The displacements are interpolated by Lagrange or Serendipity shape functions.

This local membrane surface coordinate system is defined by the boundary coordinate system (t1, t2, n).

Many features, such as an edge load, allow input in an edge local coordinate system. The orthogonal local edge coordinate system directions xl, yl, and zl are defined so that:

The local edge system can be visualized by plotting the components of the local edge transformation matrix with an Arrow Line plot. The matrix components are defined per feature. For instance, the variable name for the xx-component is <interface>.<feature_tag>.TleXX.

When an edge is shared between two or more boundaries, the directions may not always be unique. It is then possible to use the control Face Defining the Local Orientations to select from which boundary the normal direction zl should be picked. The default is Use face with lowest number.

If the geometry selection contains several edges, the only available option is Use face with lowest number, since the list of adjacent boundaries would then be different for each edge. For each edge in the selection, the face with the lowest number attached to that edge is then used for the definition of the normal orientation.

The deformation gradient F is in general defined as the gradient of the current coordinates with respect to the original coordinates:

Here is a displacement gradient computed using the tangential derivative operator, and N is the normal vector to the undeformed membrane. FT now contains information about the stretching in the plane of the membrane.

Since the tangential deformation gradient does not contain any information about the transversal stretch λn, it must be augmented by the normal deformation gradient FN to define the full deformation gradient tensor. It is given by

where n is the normal vector to the deformed membrane. For anisotropic materials, the shear deformation gradient FS is also necessary to define the full deformation gradient tensor. It is given by

where t1 and t2 are the tangent vectors to the deformed membrane. The full deformation gradient tensor F is the sum of tangential, shear and normal deformation gradient tensors

Note that FS is only nonzero for anisotropic materials, otherwise FS = 0.

The Right Cauchy-Green tensor C is generally defined as

From C, the Green-Lagrange strains are computed using the standard expression

In the membrane, only the C tensor is available, so instead the following expression is used:

where the forces (F) can be distributed over a boundary or an edge or be concentrated in a point. In the special case of a follower load, defined by its pressure p, the force intensity is where n is the normal in the deformed configuration.